One-Dimensional Dynamical Systems

f(x) = x (1 - x)

is known as the Logistic map. The Logistic map depends on the parameter . As we already saw in the Introduction, the dynamics of the map depends on this parameter. However, the difference in the dynamics is not qualitatively significant for every different . In dynamical systems theory, we like to think of the whole set of maps depending on as one object. To express this idea, we speak of the one-parameter Logistic family. We discuss the dynamics of such a family by showing the qualitatively different types of possible dynamics, and by showing in which intervals of the parameter space these dynamics occur. The one-parameter Logistic family is our object of study.

Visualising iterations

• Iterates versus time
  
• Graphical iteration
  

The dependency on and x is nicely illustrated by the Java Applet Iteration of the Logistic Map of Andy Burbanks. Use this applet to study how the graph of the map changes as you increase , and what happens with the orbit of the chosen initial point. For fixed do the orbits of all the points in the interval [0,1] behave the same?